Spectral Reciprocity for the first moment of triple product L -functions and applications

Let F be a number field with the adele ring AF; π ₁, π ₂ be two fixed unitary cuspidal automorphic representations of PGL₂(AF) with finite coprime conductors u and v, respectively; and q,l be two coprime integral ideals with (ql, uv)=1. Following the work of R. Zacharias, we estimate the first moment of L(1/2, π ⊗ π ₁ ⊗ π ₂) twisted by the Hecke eigenvalues λ π(l), where π runs through unitary automorphic representations with finite conductors dividing uvq. By applying the triple product integrals, spectral decomposition and the Plancherel formula, we get a reciprocity formula that links the twisted first moment of triple product L-functions to the spectral expansion of certain triple product periods over automorphic representations with finite conductors dividing l. As an application, we study the subconvexity problem for the triple product L-functions in the level aspect and give a subconvexity bound for L(1/2, π ⊗ π ₁ ⊗ π ₂) in terms of the norm of q.